The Van der Pol oscillator is a second-order nonlinear autonomous ODE that describes a self-sustaining limit-cycle oscillator with nonlinear damping. First studied by Balthasar van der Pol while analyzing triode circuits, it became a fundamental model in nonlinear dynamics with applications in neuroscience (FitzHugh-Nagumo nerve-impulse model), cardiac rhythms, climate oscillations, and many biological periodic phenomena.
The defining property is the limit cycle: for any initial conditions (x₀, v₀), the phase-space trajectory converges to a unique closed orbit. The nonlinearity parameter μ controls cycle shape — small μ yields near-sinusoidal oscillation (nearly circular phase portrait), while large μ produces a relaxation oscillator: slow drift along cubic nullclines punctuated by fast switches. This transition from smooth oscillation to stiff relaxation is one of the most visually striking bifurcation phenomena in all of ODE theory.
This interactive model integrates the Van der Pol equations numerically and displays both the position-versus-time chart and the phase portrait simultaneously. You can vary μ, initial conditions, and the RK4 step size to watch how trajectories from widely different starting points all converge to the same limit cycle, how the cycle's shape deforms as μ increases, and how numerical accuracy degrades when the step size becomes too large for the stiff regime.